5- Numerical results

We present AMR computations on some examples from [2] to illustrate that source terms, non conservative hyperbolic systems, capacity form differencing, and curvilinear gridscan all be successfully handled.

Example 1. Consider a linear equation of acoustics ( a hyperbolic system of three equations) with a discontinity in the sound speed across a line oblique to the grid. A plane wave strikes the interface at some angle, leading to transmitted and reflected waves. The time-evolution is best as in the figure below. The acoustics equations for the pressure perturbation p and velocities u and v can be written

where

The coefficients are the density rho(x,y) and bulk modulus of elasticity K(x,y). In the example rho has a discontinuity across the interface while K is constant. The Riemann solvers for this system in the wave-propagation for are given in [2].

Figure 1(a) shows a contour plot of the initial pressure, a cosine hump moving towards the upper right. The dashed line shows thz location of the discontinuity in sound speed. the heterogeneous material is described by a density and bulk modolus of elasticity, and here the bulk modulus is taken to be constant while the density is discontinuous, leading to the discontinuity in speed sound.

Figure 1

Example 2. The previous example does not fully test the new interface condition between the fine and coarse grids in the non-conservative case. These acoustics equations fail to be in conservation form only along the interface where the density is continuous, and the wave stays embedded in Level 3 grids as it moves along this interface. As a more severe test, computation is repeated with a simple change in the error estimation procedure so that points are flagged for refinement only if x < 0.6. For x > 0.6 there is only the coarsest grid, so the wave moves from the initial fine grids onto the coarse grid as time advances. Figure 1 shows a sequence of times ending with the time shown in Figure 2. Some smearing of the wave is seen on the coarser grid, which is inevitable, but no difficulties are observed along the discontinuity in density.

Figure 2